I want to solve a nonlinear optimization problem of the following form
\begin{equation} \min\left(\sum_i d^{x_i}c_{i}\right)\\ 0 \leq x_{i} \leq a\\ \sum_{i} x_{i} \leq b \end{equation}
$a$, $b$, $0.95 < d < 1$ and $c_i > 1$ are constant. The problem has a very specific structure, and my question is whether using a very general nonlinear optimization software (e.g. NLopt, Ipopt) is the best way to solve it numerically. The second issue is that the number of variables ($x_i$) is very large (around one million). Does the standard algorithms for nonlinear optimization can handle this large number of variables?